We introduce a fully-corrective generalized conditional gradient method for
convex minimization problems involving total variation regularization on
multidimensional domains. It relies on alternating between updating an active
set of subsets of the spatial domain as well as of an iterate given by a conic
combination of the associated characteristic functions. Different to previous
approaches in the same spirit, the computation of a new candidate set only
requires the solution of one prescribed mean curvature problem instead of the
resolution of a fractional minimization task analogous to finding a generalized
Cheeger set. After discretization, the former can be realized by a single run
of a graph cut algorithm leading to significant speedup in practice. We prove
the global sublinear convergence of the resulting method, under mild
assumptions, and its asymptotic linear convergence in a more restrictive
two-dimensional setting which uses results of stability of surfaces of
prescribed curvature under perturbations of the curvature. Finally, we
numerically demonstrate this convergence behavior in some model PDE-constrained
minimization problems.

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